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5 Reflectivity and amplitude variations with offset (AVO) in isotropic media

5 Reflectivity and amplitude variations with offset (AVO) in isotropic media

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97



3.5 Reflectivity and AVO in isotropic media



r1Ј V1

r2Ј V2



Figure 3.5.1 Reflection of a normal-incidence wave at an interface between two thick

homogeneous, isotropic, elastic layers.



IP ¼ VP



IS ¼ VS



where IP , IS are P- and S-wave impedances, VP , VS are P- and S-wave velocities, and

r is density.

At an interface between two thick homogeneous, isotropic, elastic layers, the

normal incidence reflectivity, defined as the ratio of the reflected wave amplitude

to the incident wave amplitude, is

RPP ¼

%

RSS ¼

%



2 VP2 À 1 VP1 IP2 IP1



2 VP2 ỵ 1 VP1 IP2 ỵ IP1

1

lnIP2 =IP1 ị

2

2 VS2 1 VS1 IS2 IS1



2 VS2 ỵ 1 VS1 IS2 ỵ IS1

1

lnIS2 =IS1 ị

2



where RPP is the normal incidence P-to-P reflectivity, RSS is the S-to-S reflectivity,

and the subscripts 1 and 2 refer to the first and second media, respectively

(Figure 3.5.1). The logarithmic approximation is reasonable for |R| < 0.5 (Castagna,

1993). A normally incident P-wave generates only reflected and transmitted P-waves.

A normally incident S-wave generates only reflected and transmitted S-waves. There

is no mode conversion.



AVO: amplitude variations with offset

For non-normal incidence, the situation is more complicated. An incident P-wave

generates reflected P- and S-waves and transmitted P- and S-waves. The reflection



98



Seismic wave propagation



VP1, VS1, r1



qS1



Reflected

S-wave



q1



Reflected

P-wave



Incident

P-wave



Transmitted

P-wave

q2



qS2



Transmitted

S-wave



VP2, VS2, r2



Figure 3.5.2 The angles of the incident, reflected, and transmitted rays of a P-wave

with non-normal incidence.



and transmission coefficients depend on the angle of incidence as well as on the

material properties of the two layers. An excellent review is given by Castagna

(1993).

The angles of the incident, reflected, and transmitted rays (Figure 3.5.2) are related

by Snell’s law as follows:





sin 1 sin 2 sin S1 sin S2

¼

¼

¼

VP1

VP2

VS1

VS2



where p is the ray parameter. y and yS are the angles of P- and S-wave propagation,

respectively, relative to the reflector normal. Subscripts 1 and 2 indicate angles or

material properties of layers 1 and 2, respectively.

The complete solution for the amplitudes of transmitted and reflected P- and

S-waves for both incident P- and S-waves is given by the Knott–Zoeppritz equations

(Knott, 1899; Zoeppritz, 1919; Aki and Richards, 1980; Castagna, 1993).

Aki and Richards (1980) give the results in the following convenient matrix form:

0

1

#"

""

#"

""

B PP SP PP SP C

B #" #" "" "" C

B PS SS PS SS C

B

C

B ## ## "# "# C ¼ MÀ1 N

B

C

B PP SP PP SP C

@

A

"#



"#



PS SS PS



##



##



SS



where each matrix element is a reflection or transmission coefficient for displacement

amplitudes. The first letter designates the type of incident wave, and the second letter

designates the type of reflected or transmitted wave. The arrows indicate downward #

and upward ↑ propagation, so that a combination #" indicates a reflection coefficient,

while a combination ## indicates a transmission coefficient. The matrices M and N

are given by



99



3.5 Reflectivity and AVO in isotropic media

0



À cos S1



sin 2



À sin S1



cos 2



À

Á

1 VS1 1 À 2 sin2 S1



22 VS2 sin S2 cos 2



1 VS1 sin 2S1



À

Á

2 VP2 1 À 2 sin2 S2



À sin 1



B

B

B

cos 1

B

M ¼B

B

B 21 VS1 sin S1 cos 1

B

@

À

Á

À1 VP1 1 À 2 sin2 S1

0



sin 1

B

B

B

cos 1

B

N ¼B

B

B 21 VS1 sin S1 cos 1

B

@

À

Á

1 VP1 1 À 2 sin2 S1



cos S2



1



C

C

C

C

C

À

Á

C

2 VS2 1 À 2 sin2 S2 C

C

A

À2 VS2 sin 2S2

À sin S2



cos S1



À sin 2



À cos S2



À sin S1



cos 2



À

Á

1 VS1 1 À 2 sin2 S1



22 VS2 sin S2 cos 2



À1 VS1 sin 2S1



À

Á

À2 VP2 1 À 2 sin2 S2



1



C

C

C

C

C

À

Á

C

2 VS2 1 À 2 sin2 S2 C

C

A

2 VS2 sin 2S2

À sin S2



Results for incident P and incident S, given explicitly by Aki and Richards (1980),

#"



#"



##



##



#"



#"



are as follows, where RPP ¼ PP; RPS ¼ PS; TPP ¼ PP; TPS ¼ PS; RSS ¼ SS; RSP ¼ SP;

##



##



TSP ¼ SP; TSS ¼ SS:



RPP











!.

cos 1

cos 2

cos 1 cos S2

2

ẳ b

c

F aỵd

Hp

D

VP1

VP2

VP1 VS2



RPS ẳ 2







!.

cos 1

cos 2 cos S2

ab ỵ cd

pVP1 VS1 Dị

VP1

VP2 VS2



TPP ẳ 21



cos 1

FVP1 =VP2 Dị

VP1



TPS ¼ 21



cos 1

HpVP1 =ðVS2 DÞ

VP1



RSS ¼ À











!.

cos S1

cos S2

cos 2 cos S1

E aỵd

Gp2 D

b

c

VS1

VS2

VP2 VS1



RSP







cos S1

cos 2 cos S2

pVS1 =VP1 Dị

ẳ2

ab ỵ cd

VS1

VP2 VS2



TSP ẳ 21

TSS ẳ 21



cos S1

GpVS1 =VP2 Dị

VS1



cos S1

EVS1 =VS2 Dị

VS1



100



Seismic wave propagation



where









a ẳ 2 1 À 2 sin2 S2 À 1 1 À 2 sin2 S1





b ẳ 2 1 2 sin2 S2 ỵ 21 sin2 S1





c ẳ 1 1 2 sin2 S1 ỵ 22 sin2 S2

À

Á

2

2

d ¼ 2 2 VS2

À 1 VS1

D ¼ EF ỵ GHp2 ẳ det Mị=VP1 VP2 VS1 VS2 ị

cos 1

cos 2

ỵc

VP1

VP2

cos S1

cos S2

Fẳb

ỵc

VS1

VS2

cos 1 cos S2

Gẳad

VP1 VS2

cos 2 cos S1

H ¼aÀd

VP2 VS1

E¼b



and p is the ray parameter.



Approximate forms

Although the complete Zoeppritz equations can be evaluated numerically, it is often

useful and more insightful to use one of the simpler approximations.

Bortfeld (1961) linearized the Zoeppritz equations by assuming small contrasts

between layer properties as follows:



 



!

1

VP2 2 cos 1

sin 1 2 2

lnð2 =1 ị

2

RPP 1 ị % ln



VS1 VS2

ị 2ỵ

VP1 1 cos 2

VP1

2

lnðVS2 =VS1 Þ

Aki and Richards (1980) also derived a simplified form by assuming small layer

contrasts. The results are conveniently expressed in terms of contrasts in VP, VS, and

r as follows:

RPP ị %





1

1 VP

VS



4p2 V"S2 "

1 4p2 V"S2

2

"

"

2

2 cos  VP

VS



ÀpV"P

RPS ðÞ %

2 cos S









2 2

2 cos  cos S 

"

"

1 2VS p ỵ 2VS "

"

VP V"S







!

2 "2

2 cos  cos S ÁVS

"

À 4p VS À 4VS "

VP V"S

V"S





1 

1

VP



1

TPP ị % 1

2

2 "

2 cos 

V"P



101



3.5 Reflectivity and AVO in isotropic media



pV"P

TPS ðÞ %

2 cos S







2 2

2 cos  cos S Á

"

"

1 À 2VS p À 2VS "

"

VP V"S







!

2 "2

2 cos  cos S ÁVS

"

À 4p VS ỵ 4VS "

VP V"S

V"S

RSP ị%



cos S V"S

RPS ị

V"P cos 







Á



1

2 "2 Á

2 "2 ÁVS

À

À 4p VS

RSS ðÞ% À 1 À 4p VS

"

2

2 cos2 S

V"S

TSP ðÞ % À



cos S V"S

TPS

V"P cos 







1 

1

VS



TSS ị % 1

1

2

2 "

2 cos S

V"S

where

pẳ



sin 1 sin S1



VP1

VS1



 ẳ 2 ỵ 1 ị=2

S ẳ S2 ỵ S1 ị=2



 ẳ 2 1



" ẳ 2 ỵ 1 ị=2



VP ẳ VP2 VP1



V"P ẳ VP2 ỵ VP1 ị=2



VS ẳ VS2 VS1



V"S ẳ VS2 ỵ VS1 Þ=2



Often, the mean P-wave angle y is approximated as y1, the P-wave angle of

incidence.

The result for P-wave reflectivity can be rewritten in the familiar form:

RPP ị % RP0 ỵ B sin2  ỵ Ctan2  sin2 ị

or







!

1 VP 

1 VP

VS 

V"S2



2 "2 2 " ỵ



sin2 

RPP ị %

2 V"P

"

2 V"P

"

VP

VS







1 VP 2

2

tan





sin



2 V"P



This form can be interpreted in terms of different angular ranges (Castagna, 1993).

In the above equations RP0 is the normal incidence reflection coefficient as

expressed by



102



Seismic wave propagation



RP0 ẳ







IP2 IP1 IP 1 VP 

%

%



"

IP2 ỵ IP1

2IP

2 V"P



The parameter B describes the variation at intermediate offsets and is often called the

AVO gradient, and C dominates at far offsets near the critical angle.

Shuey (1985) presented a similar approximation where the AVO gradient is

expressed in terms of the Poisson ratio n as follows:

"

#

Á

Án

1 ÁVP À 2

2

RPP ð1 Þ % RP0 ỵ ERP0 ỵ

tan





sin



sin2 1 ỵ

1

1

2

2 V"P

1 "nị

where





1 VP 



RP0 %

"

2 V"P





1 2"n

E ẳ F 21 ỵ Fị

1 "n

Fẳ



VP =V"P

VP =V"P ỵ ="





and

n ẳ n2 n1

"n ẳ n2 ỵ n1 ị=2

The coefficients E and F used here in Shuey’s equation are not the same as those

defined earlier in the solutions to the Zoeppritz equations.

Smith and Gidlow (1987) offered a further simplification to the Aki–Richards

equation by removing the dependence on density using Gardner’s equation (see

Section 7.10) as follows:

 / V 1=4

giving

ÁVP

ÁVS

RPP ðÞ % c " þ d "

VP

VS

where

5 1 V"2

1

c ¼ À "S2 sin2  þ tan2 

8 2 VP

2

V"2

d ¼ À 4 "S2 sin2 

VP



103



3.5 Reflectivity and AVO in isotropic media



Wiggins et al. (1983) showed that when VP % 2VS, the AVO gradient is approximately (Spratt et al., 1993)

B % RP0 À 2RS0

given that the P and S normal incident reflection coefficients are





1 ÁVP 



RP0 %

"

2 V"P





1 VS 

RS0 %



"

2 V"S

Hilterman (1989) suggested the following slightly modified form:

RPP ị % RP0 cos2  ỵ PR sin2 

where RP0 is the normal incidence reflection coefficient and

PR ẳ



n2 n1

1 "nị2



This modified form has the interpretation that the near-offset traces reveal the P-wave

impedance, and the intermediate-offset traces image contrasts in Poisson ratio

(Castagna, 1993).

Gray et al. (1999) derived linearized expressions for P–P reflectivity in terms of the

angle of incidence, y, and the contrast in bulk modulus, K, shear modulus, m, and bulk

density, r:





 2 



V"S

1 1 V"S2 2 K

1 2



2

Rị ẳ "2 sec 

ỵ "2

sec  2 sin 

4 3 VP

K

3



VP







1 1 2





sec 

2 4



Similarly, their expression in terms of Lames coefficient, l, shear modulus, and bulk

density is





 2 



V"S

1 1 V"S2 2 l

1 2



2

ỵ "2

sec  2 sin 

Rị ẳ "2 sec 

4 2 VP

l

2



VP









1 1 2

Á

À sec 

2 4





In the same assumptions of small layer contrast and limited angle of incidence, we

can write the linearized SV-to-SV reflection (Ruăger, 2001):





1 IS

7 VS



1 VS 2

RSV-iso S ị % " ỵ

ỵ2

sin S tan2 S

sin2 S

"

"

2 IS

2 VS

2 V"S



104



Seismic wave propagation



where yS is the SV-wave phase angle of incidence and IS ¼ VS is the shear

impedance.

Similarly, for SH-to-SH reflection (Ruăger, 2001):

RSH S ị ẳ







1 IS 1 VS

tan2 S



2 I"S

2 V"S



where yS is the SH-wave phase angle of incidence.

For P-to-SV converted shear-wave reflection (Aki and Richards, 1980):

0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

 2

À tan S @

Á

V"S

2

À sin2 S A

RPS ðS Þ % " " 1 2 sin S ỵ 2 cos S

"

"

2VS =VP

VP

0



s 1

 2

V"S

tan S

VS

ỵ " " @4 sin2 S À 4 cos S

À sin2 S A "

V"P

VS

2VS =VP

where yS is the S-wave phase angle of reflection. This can be rewritten as (Gonza´lez,

2006)

RPS ðÞ %



À sin 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

ðV"P =V"S Þ ðV"P =V"S Þ À sin2 

82

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3

 2

< 1 V" 2

V"P



P

2

4



sin





cos



sin2  5

"

"

: 2 VS

"

VS

9

s 3

 2

=

"

VP

2 5 ÁVS

À42 sin2  À 2 cos 

À

sin



V"S

V"S ;

2



where y is the P-wave phase angle of incidence.

For small angles, Duffaut et al. (2000) give the following expression for RPS(y):





!

V"S Á

V"S ÁVS

1

À 2 " " sin 

RPS ị %

1ỵ2 "

2

VP "

VP VS

V"S

ỵ "

VP



!







1 V"S 

V"S 1  2VS

ỵ "

sin3 

"



"

4 VP "

V"P 2

VS



which can be simplified further (Jı´lek, 2002b) to





!

1

V"S 

V"S VS

RPS ị %

1ỵ2 "

2 " " sin 

2

VP 

VP VS



105



3.6 Plane-wave reflectivity in anisotropic media



Assumptions and limitations

The equations presented in this section apply in the following cases:

 the rock is linear, isotropic, and elastic;

 plane-wave propagation is assumed; and

 most of the simplified forms assume small contrasts in material properties across the

boundary and angles of incidence of less than about 30 . The simplified form for

P-to-S reflection given by Gonza´lez (2006) is valid for large angles of incidence.



3.6



Plane-wave reflectivity in anisotropic media

Synopsis

An incident wave at a boundary between two anisotropic media (Figure 3.6.1) can

generate reflected quasi-P-waves and quasi-S-waves as well as transmitted quasiP-waves and quasi-S-waves (Auld, 1990). In general, the reflection and transmission

coefficients vary with offset and azimuth. The AVOA (amplitude variation with

offset and azimuth) can be detected by three-dimensional seismic surveys and is a

useful seismic attribute for reservoir characterization.

Brute-force modeling of AVOA by solving the Zoeppritz (1919) equations can be

complicated and unintuitive for several reasons: for anisotropic media in general, the

two shear waves are separate (shear-wave birefringence); the slowness surfaces are

nonspherical and are not necessarily convex; and the polarization vectors are neither

parallel nor perpendicular to the propagation vectors.

Schoenberg and Prota´zio (1992) give explicit solutions for the plane-wave reflection and transmission problem in terms of submatrices of the coefficient matrix of the

Zoeppritz equations. The most general case of the explicit solutions is applicable to

Reflected

qS-wave



Incident qP-wave



Reflected

qP-wave



q1



r1, a1, b1, e1, d1, g1

r2, a2, b2, e2, d2, g2

q2



Transmitted

qP-wave



Transmitted

qS-wave



Figure 3.6.1 Reflected and transmitted rays caused by a P-wave incident at a boundary between two

anisotropic media.



106



Seismic wave propagation



monoclinic media with a mirror plane of symmetry parallel to the reflecting plane.

Let R and T represent the reflection and transmission matrices, respectively,

2



RPP



6

R ¼4 RPS

RPT

2



TPP

T ¼ 4 TPS

TPT



RSP



RTP



3



RSS



7

RTS 5



RST



RTT



TSP

TSS

TST



3

TTP

TTS 5

TTT



where the first subscript denotes the type of incident wave and the second subscript

denotes the type of reflected or transmitted wave. For “weakly” anisotropic media, the

subscript P denotes the P-wave, S denotes one quasi-S-wave, and T denotes the other

quasi-S-wave (i.e., the tertiary or third wave). As a convention for real s23P , s23S , and s23T ,

s23P < s23S < s23T

where s3i is the vertical component of the phase slowness of the ith wave type when

the reflecting plane is horizontal. An imaginary value for any of the vertical slownesses implies that the corresponding wave is inhomogeneous or evanescent. The

impedance matrices are defined as

2

3

eP1

eS1

eT1

6

7

eP2

eS2

eT2

6

7

6

7

X ¼6 fÀðC13 eP1 þ C36 eP2 Þs1 fÀðC13 eS1 þ C36 eS2 Þs1 fC13 eT1 ỵ C36 eT2 ịs1 7

6

7

4 C23 eP2 ỵ C36 eP1 ịs2

C23 eS2 ỵ C36 eS1 ịs2

C23 eT2 ỵ C36 eT1 ịs2 5

C33 eP3 s3P g

2



fC55 s1 ỵ C45 s2 ịeP3

6 C55 eP1 ỵ C45 eP2 ịs3P g

6

6

6

Y ẳ 6 fC s ỵ C s ịe

45 1

44 2 P3

6

6 C e ỵ C e ịs g

45 P1

44 P2 3P

4

eP3



C33 eS3 s3S g

fC55 s1 ỵ C45 s2 ịeS3

C55 eS1 ỵ C45 eS2 ịs3S g

fC45 s1 ỵ C44 s2 ịeS3

C45 eS1 þ C44 eS2 Þs3S g

eS3



ÀC33 eT3 s3T g

3

fÀðC55 s1 þ C45 s2 ịeT3

C55 eT1 ỵ C45 eT2 ịs3T g 7

7

7

7

fC45 s1 ỵ C44 s2 ịeT3 7

7

C45 eT1 ỵ C44 eT2 Þs3T g 7

5

eT3



where s1 and s2 are the horizontal components of the phase slowness vector; eP, eS,

and eT are the associated eigenvectors evaluated from the Christoffel equations (see

Section 3.2), and CIJ denotes elements of the stiffness matrix of the incident medium.

X0 and Y0 are the same as above except that primed parameters (transmission

medium) replace unprimed parameters (incidence medium). When neither X nor Y

is singular and (X1X0 ỵ Y1Y0 ) is invertible, the reflection and transmission coefficients can be written as



107



3.6 Plane-wave reflectivity in anisotropic media



T ẳ 2X1 X0 ỵ Y1 Y0 ị1

R ẳ X1 X0 Y1 Y0 ịX1 X0 þ YÀ1 Y0 ÞÀ1

Schoenberg and Prota´zio (1992) point out that a singularity occurs at a horizontal

slowness for which an interface wave (e.g., a Stoneley wave) exists. When Y is

singular, straightforward matrix manipulations yield

T ẳ 2Y01 YX1 X0 Y01 Y ỵ Iị1

R ẳ X1 X0 Y01 Y ỵ Iị1 X1 X0 Y0À1 Y À IÞ

Similarly, T and R can also be written without X–1 when X is singular as

T ¼ 2X0À1 XI ỵ Y1 Y0 X01 Xị1

R ẳ I ỵ Y1 Y0 X0À1 XÞÀ1 ðI À YÀ1 Y0 X0À1 XÞ

Alternative solutions can be found by assuming that X0 and Y0 are invertible

R ẳ Y01 Y ỵ X01 Xị1 Y01 Y X01 Xị

T ẳ 2X01 XY01 Y ỵ X01 Xị1 Y01 Y

ẳ 2Y01 YY01 Y ỵ X01 Xị1 X01 X

These formulas allow more straightforward calculations when the media have at

least monoclinic symmetry with a horizontal symmetry plane.

For a wave traveling in anisotropic media, there will generally be out-of-plane

motion unless the wave path is in a symmetry plane. These symmetry planes include

all vertical planes in VTI (transversely isotropic with vertical symmetry axis) media

and the symmetry planes in HTI (transversely isotropic with horizontal symmetry

axis) and orthorhombic media. In this case, the quasi-P- and the quasi-S-waves in the

symmetry plane uncouple from the quasi-S-wave polarized transversely to the

symmetry plane. For weakly anisotropic media, we can use simple analytical formulas

(Banik, 1987; Thomsen, 1993; Chen, 1995; Ruăger, 1995, 1996) to compute AVOA

(amplitude variation with offset and azimuth) responses at the interface of anisotropic

media that can be either VTI, HTI, or orthorhombic. The analytical formulas give

more insight into the dependence of AVOA on anisotropy. Vavrycˇuk and Psˇencˇ´ık

(1998) and Psˇencˇ´ık and Martins (2001) provide formulas for arbitrary weak

anisotropy.



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